Algebra

Linear algebra lecture at Helsinki University of Technology

This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation).

Algebra (from Arabic: الجبر, al-ğabr) is a branch of mathematics concerning the study of structure, relation and quantity. Elementary algebra is often taught in high school and gives an introduction to the basic ideas of algebra: studying what happens when numbers are added or multiplied, and how to make polynomials and find their roots.

Algebra is much broader than arithmetic and can be generalized. Rather than working directly with numbers, one can work with symbols or elements of some set. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

Together with geometry and analysis, algebra is one of the three main branches of mathematics.

Classification

Algebra may be roughly divided into the following categories:

• elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
• abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
• linear algebra, in which the specific properties of vector spaces are studied (including matrices);
• universal algebra, in which properties common to all algebraic structures are studied.

In advanced studies, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:

• Normed linear spaces
• Banach spaces
• Hilbert spaces
• Banach algebras
• Normed algebras
• Topological algebras
• Topological groups

Elementary algebra

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. Although in arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, numbers are often denoted by symbols (such as a, x, y). This is useful because:

• It allows the general formulation of arithmetical laws (such as $a+b=b+a$ for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
• It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that $3x+1=10$").
• It allows the formulation of functional relationships (such as "If you sell x tickets, then your profit will be $3x-10$ dollars, or $f(x)=3x-10$, where f is the function, and x is the number the function is performed on.").

Abstract algebra

Main article: Abstract algebra; see also algebraic structures.

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of objects called elements. All the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax²+bx+c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is abstracted to give a binary operation, * say. For two elements a and b in a set S a*b gives another element in the set, (technically this condition is called closure). Addition (+), subtraction (-), multiplication (×), and division (÷) are all binary operations as in addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator * the identity element e must satisfy a*e=a and e*a=a. This holds for addition as a+0=a, and 0+a=a and multiplication a×1=a, 1×a=a. However, if we take the positive natural numbers and addition, there is no identity element.

Inverse elements: The negative numbers gives rise to the concept of an inverse elements. For addition, the inverse of a is -a, and for multiplication the inverse is ¹/_a. A general inverse element a^-1 must satisfy the property that a*a^-1=e and a^-1*a=e.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2+3)+4=2+(3+4). In general, this becomes (a+b)+c=a+(b+c). This property is shared by most binary operations, but not subtraction or division.

Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes a+b=b+a. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication.

Groups

Main article: group; see also group theory, examples of groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group consists of:

• a set S of elements,
• a (closed) binary operation (*)
• an identity element exists,
• every element has an inverse,
• the operation is associative.

If commutativity is included as well, then we get an Abelian group.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, -a. The associativity requirement is met, because for any integers a, b and c, $(a+b)+c=a+(b+c)$.

The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since $1\cdot a=a\cdot 1=a$ for any rational number a. The inverse of a is ${\frac {1}{a}}$, since $a\cdot {1 \over a}=1$.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¹/₄, which is not an integer.

The theory of groups is studied in group theory. A major result in this theory is the Classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.

Examples of groups
Set:	Natural numbers $\mathbb {N}$		Integers $\mathbb {Z}$		Rational numbers $\mathbb {Q}$ (also real $\mathbb {R}$ and complex $\mathbb {C}$ numbers)				Integers mod 3: {0,1,2}
operation	+ (including zero)	× (excluding zero)	+	× (excluding zero)	+	−	× (excluding zero)	÷ (excluding zero)	+
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
identity	0	1	0	1	0	NA	1	NA	0
inverse	NA	NA	-1	NA	-1	NA	1/a	NA	0,2,1 respectively
Associative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes
Commutative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes
Structure	semigroup	quasigroup	Abelian group	Monoid	Abelian group	quasigroup	Abelian group	quasigroup	Abelian group

Many other types of algebraic structures exist. Among the most common are rings, fields, and monoids. These different structures can be used to model different types of mathematical objects. Different algebraic structures are often related. For example, a group is a specific kind of monoid, and rings and fields are similar to groups, but with more operations.

Algebras

The word algebra is also used for various algebraic structures:

• algebra over a field
• algebra over a set
• Boolean algebra
• F-algebra and F-coalgebra in category theory
• sigma-algebra

History

Hellenistic mathematician Euclid details geometrical algebra in Elements.

The origins of algebra can be traced to the ancient Babylonians, who used an early type of algebra to solve linear, quadratic, and indeterminate linear equations in the second millenium BC. By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC usually solved such equations by geometric methods, such as those described in the Moscow and Rhind Mathematical Papyri, Sulba Sutras, Euclid's Elements, and The Nine Chapters on the Mathematical Art.

Indian mathematicians proceeded to write treatises on algebraic means of solving equations from the end of the first millenium BC, followed by Hellenistic mathematicians from the early first millennium AD. Important algebraic works from this general era include the Bakhshali Manuscript, the works of Hero of Alexandria, the Arithmetica of Diophantus, the Aryabhatiya of Aryabhata, and the Brahma Sputa Siddhanta of Brahmagupta.

The word "algebra" is named after the Arabic word "al-jabr" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by Muḥammad ibn Mūsā al-Ḵwārizmī in 820. The word al-jabr means "reunion". Al-Khwarizmi is often considered the "father of algebra" (though that title is also given to Diophantus), as much of his works on reduction are still in use today. Another Persian mathematician Omar Khayyam developed algebraic geometry and found the geometric solution of the general cubic equation, while Indian mathematician Bhaskara and Chinese mathematician Zhu Shijie solved some higher-order equations.

A key event in the further development of algebra was the algebraic solution of the general cubic and quartic equations (biquadratic equations), developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.

The stages of the development of symbolic algebra are roughly as follows:

• rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century;
• geometric constructive algebra, which was emphasised by the Vedic Indian and classical Greek mathematicians;
• syncopated algebra, as developed by Diophantus and in the Bakhshali Manuscript; and
• symbolic algebra, which sees its culmination in the work of Leibniz.

Cover of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

A timeline of key algebraic developments are as follows:

• Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.
• Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.
• Circa 800 BC: Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax² = c and ax² + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.
• Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.
• Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.
• Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.
• Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.
• Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.
• Circa 150 AD: Hellenized Egyptian mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.
• Circa 200: Hellenized Babylonian mathematician Diophantus, who lived in Egypt and is often considered the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
• 499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a differential equation.
• Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.
• 628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell's equation, and gives rules for solving linear and quadratic equations. He discovers that quadratic equations have two roots, including both negative as well as irrational roots.
• 820: The word algebra is derived from operations described in the treatise written by the Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.
• Circa 990: Persian mathematician Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities.
• Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations.
• 1072: Persian mathematician Omar Khayyam develops algebraic geometry and, in the Treatise on Demonstration of Problems of Algebra, gives a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
• 1114: Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves quadratic equations with more than one unknown, some cubic, quartic and higher-order polynomial equations, Pell's equation, the general indeterminate quadratic equation, aswell as some indeterminate cubic, quartic and higher-order equations.
• 1150: Bhaskara, in his Siddhanta Shiromani, solves differential equations.
• 1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.
• Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with upto four unknowns, and numerically solves some quartic and higher-order polynomial equations.
• Circa 1400: Indian mathematician Madhava of Sangamagramma finds the solution of transcendental equations by iteration, iterative methods for the solution of non-linear equations, and solutions of differential equations.
• 1515: Scipione del Ferro solves a cubic such that the quadratic term is missing.
• 1535: Nicolo Fontana Tartaglia solves a cubic such that the linear term is missing.
• 1545: Girolamo Cardano publishes Ars magna -The great art which gives solutions for a variety of cubics as well as Ludovico Ferrari's solution of a special quartic equation.
• 1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.
• 1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.
• 1631: Thomas Harriot in a posthumus publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than".
• 1682: Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.
• 1683: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, discriminant, and Bernoulli numbers.
• 1685: Kowa Seki solves the general cubic equation, aswell as some quartic and fifth-order polynomial equations.
• 1693: Leibniz solves systems of simultaneous linear equations using matrices and determinants.
• 1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.
• 1830: Galois theory is developed by Évariste Galois in his work on abstract algebra.

References

• Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
• Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
• George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Penguin Books, 2000).
• John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005).

See also

Algebra

• Fundamental theorem of algebra (which is really a theorem of mathematical analysis, not of algebra)
• Computer algebra system

External links

• Explanation of Basic Topics
• Sparknotes' Review of Algebra I and II
• Understanding Algebra. An online algebra text by James W. Brennan.
• Algebra--the basic ideas     First of 6 parts in a short course on basic algebra at the high school level.
• Highlights in the history of algebra
• ExampleProblems.com Example problems and solutions from basic and abstract algebra.